Polar Quantization Research Articles

Multidimensional rotations for robust quantization of image data

Laplacian and generalized Gaussian data arise in the transform and subband coding of images. This paper describes a method of rotating independent, identically distributed (i.i.d.) Laplacian-like data in multiple dimensions to significantly improve the overload characteristics for quantization. The rotation is motivated by the geometry of the Laplacian probability distribution, and can be achieved with only additions and subtractions using a Walsh-Hadamard transform. Its theoretical and simulated results for scalar, lattice, and polar quantization are presented in this paper, followed by a direct application to image compression. We show that rotating the image data before quantization not only improves compression performance, but also increases robustness to the channel noise and deep fades often encountered in wireless communication.

A vector quantizer for the Laplace source

The low complexity, nearly optimal vector quantizer (VQ) is a generalization of T. R. Fischer's (1986) pyramid VQ and is similar in structure to the unrestricted polar quantizers previously presented for the independent Gaussian source. An analysis of performance is presented with results for both the product code pyramid VQ and the unrestricted version. This analysis, although asymptotic in nature, helps to demonstrate the performance advantages of the VQ. Implementation issues of the VQ are discussed. Nonasymptotic results are considered. In particular, the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this VQ through several example designs with Monte Carlo simulations of performance. For the restricted form (the pyramid VQ), the author provides further implementational information and low dimension analytical results. >

Polar quantizing for coded PSK transmission

The issue of demodulator quantization prior to decoding in error control systems employing M-ary phase-shift keying (PSK) as a modulation technique is considered. The use of angle-only, or polar, quantization with J>or=M levels, which is an easily implemented alternative to rectangular quantization, is described. The R/sub 0/ parameter is presented as an effective indicator of coding potential for quantized M-ary PSK channels, and different designs in the R/sub 0/ context are discussed. The principal results are that for J=2M, a polar quantizer satisfies the necessary condition for optimality, and it is conjectured that the design is in fact optimal. The corresponding performance closes about half the gap (in decibels) between the hard-decision demodulator and the unquantized demodulator. Angle-only quantization, in the limit of fine precision, costs only a fraction of a decibel in performance for typically operated coded 8-PSK and 16-PSK. Numerical results for quantizers with J=2M and J=4M are presented, and the case of an added erasure zone is considered.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Discrimination of planar shapes using shape matrices

An algorithm is presented for describing and discriminating binary images of planar shapes. The descriptor is a matrix that the dimensions dependent on the maximum radius of the shape. The descriptor is obtained by a polar quantization of the shape and is independent of the shape's position, orientation, and size. The descriptors discriminate two or more shapes by measuring their degree of similarity. The shapes considered include letters of the alphabet, numerals, geometrical figures, and physical objects. The results obtained for the letters were compared to those reported by A. Goshtasby (1985). The comparison showed that the proposed algorithm provides a higher degree of discrimination.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotic performance of unrestricted polar quantizers (Corresp.)

Unrestricted polar quantizers (UPQ's) were previously introduced to overcome the weak mean-squared error performance of polar coordinate quantizers for the independent bivariate Gaussian source when the number of levels was small ( bits /dimension). An asymptotic ntean-squared error analysis of the UPQ's with a general circularly symmetric source is presented. The performance of the UPQ's is shown to be asymptotically within 0.17 dB of that of the optimum bivariate quantizer. The results are extended to dimensions greater than two. Examples for the independent Gaussian source are included.

Description and Discrimination of Planar Shapes Using Shape Matrices

A shape descriptor has been developed which can describe a shape independent of its translation, rotation, and scaling. The description is in the form of a matrix and it is obtained by a polar quantization of a shape. The quantization process takes into consideration not only a shape's outer geometry but its inner geometry as well. The descriptor is information preserving and if the quantization parameters are selected properly, it is possible to reconstruct an original shape from its description. In this technique, shape discrimination is possible by a simple EXCLUSIVE-OR operation on shape descriptions.

Asymptotic performance of Dirichlet rotated polar quantizers (Corresp.)

An upper bound to the asymptotic mean-square error performance of optimized rotated polar quantizers is presented for a circularly symmetric random input. This form of quantizer is of interest because it has a scalar implementation and has been shown, for a small number of levels with a Gaussian source, to have much better performance than either the rectangular or polar schemes previously documented.

Comparison of Rectangular and Polar Quantizers in Digital Holography

Complex amplitude encoded in any digital hologram must undergo quantization, usually in either polar or rectangular format. In this paper these two schemes are compared under the constraints and conditions inherent in digital holography. For Fourier transform holograms when the spectrum is levelled through phase coding, the rectangular format is shown to be optimal. In the absence of phase coding, and also if the amplitude spectrum has a large dynamic range, the polar format may be preferable.

Optimal circularly symmetric quantizers

Polar coordinates quantization of the bivariate Gaussian and other cir- cularly symmetric sources has already been investigated. The schemes quantize the polar coordinates representation of the random variables independently in an attempt to reduce the mean square error below that of an analogous rectangular coordinates quantizer. It has been shown for the Gaussian case that the polar quantizer outperforms the rectangular quantizer when the number of levels N is large, while for small N, the rectangular form is often better than the polar form. This paper is an investigation of the optimality of polar quantizers with the subsequent development of optimal circularly symmetric quantizers (labelled Dirichlet polar quantizers).

Magnitude/Phase Quantization of Independent Gaussian Variates

The performance of two-dimensional polar quantization of independent Gaussian variates is evaluated as a possible improvement over one-dimensional quantizers. The distortion measure is assumed to be squared error. The quantizers are a generalization of more restrictive polar quantizers analyzed previously, admitting a differing number of phase positions in each magnitude sector. Results are provided for several bit rates up to 2.5 bits/sample. The optimal design always performs as well or better than the restricted polar quantizer, and, in the cases analyzed, slightly outperforms the Max (single-sample) quantizers for a fixed number of levels.

Two-dimensional quantization of bivariate circularly symmetric densities

The problem of quantizing a two-dimensional random variable whose bivariate density has circular symmetry is considered in detail. Two quantization methods are considered, leading to polar and rectangular representations. A simple necessary and sufficient condition is derived to determine which of these two quantization schemes is best. If polar quantization is deemed best, the question arises as to the ratio of the number of phase quantizer levels to that or magnitude quantizer levels when the product of these numbers is fixed. A simple expression is derived for this ratio that depends only upon the magnitude distribution. Several examples of common circularly symmetric bivariate densities are worked out in detail using these expressions.

Polar Quantization of a Complex Gaussian Random Variable

We solve numerically the optimum fixed-level non-uniform and uniform quantization of a circularly symmetric complex (or bivariate) Gaussian random variable for the mean absolute squared error criterion. For a given number of total levels, we determine its factorization into the product of numbers of magnitude and phase levels that produces the minimum distortion. We tabulate the results for numbers of "useful" output levels up to 1024, giving their optimal factorizations, minimum distortion, and entropy. For uncoded quantizer outputs, we find that the optimal splitting of rate between magnitude and phase, averaging to 1.52 and 1.47 bits more in the phase angle than magnitude for optimum and uniform quantization, respectively, compares well with the optimal polar coding formula Of 1.376 bits of Pearlman and Gray [3]. We also compare the performance of polar to rectangular quantization by real and imaginary parts for both uncoded and coded output levels. We find that, for coded outputs, both polar quantizers are outperformed by the rectangular ones, whose distortion-rate curves nearly coincide with Pearlman and Gray's polar coding bound. For uncoded outputs, however, we determine that the polar quantizers surpass in performance their rectangular counterparts for all useful rates above 6.0 bits for both optimum and uniform quantization. Below this rate, the respective polar quantizers are either slightly inferior or comparable.